Gradient and scalar function question

[Dustinsfl]

I am trying to determine my scalar function \(f(u_1, u_2, u_3)\) of elliptical cylindrical coordinates.

\begin{align*}
x &= a\cosh(u)\cos(v)\\
y &= a\sinh(u)\sin(v)\\
z &= z
\end{align*}

I have determined my vectors \(\mathbf{U}_u\), \(\mathbf{U}_u\), and \(\mathbf{U}_z\).

\begin{align*}
\mathbf{U}_u &= a\sinh(u)\cos(v)\hat{\mathbf{i}} + a\cosh(u)\sin(v)\hat{\mathbf{j}}\\
\mathbf{U}_v &= -a\cosh(u)\sin(v)\hat{\mathbf{i}} + a\sinh(u)\cos(v)\hat{\mathbf{j}}\\
\mathbf{U}_z &= \hat{\mathbf{k}}
\end{align*}

\[
\nabla f = \frac{1}{h_1}\frac{\partial f}{\partial u_1}\hat{\mathbf{u}}_1 +
\frac{1}{h_2}\frac{\partial f}{\partial u_2}\hat{\mathbf{u}}_2 +
\frac{1}{h_3}\frac{\partial f}{\partial u_3}\hat{\mathbf{u}}_3
\]

I have found \(h_i\)'s as
\[
h_1 = h_2 = \frac{1}{a\sqrt{\cosh^2(u) - \cos^2(v)}}
\]
and
\[
h_3 = 1.
\]

So I need to find \(\frac{\partial f}{\partial u_i}\) but I don't know what my scalar funtion is.

 

[jvicens]

Can you help me find it and its partial derivatives?

Sure, I would be happy to help you with this problem. To find your scalar function \(f(u_1,u_2,u_3)\), we can use the fact that the gradient of a scalar function is perpendicular to the level surfaces of that function. In other words, the gradient points in the direction of steepest increase of the function.

Since we know the vectors \(\mathbf{U}_u\), \(\mathbf{U}_v\), and \(\mathbf{U}_z\), we can use them to construct a surface in the \(u_1u_2u_3\) coordinate system. This surface will be defined by the equation \(f(u_1,u_2,u_3)=C\), where \(C\) is a constant. We can then take the gradient of this function to find the direction of steepest increase.

To find the partial derivatives of \(f\) with respect to \(u_i\), we can use the chain rule. For example, \(\frac{\partial f}{\partial u_1}\) can be found by taking the dot product of the gradient of \(f\) with \(\mathbf{U}_u\).

\[
\frac{\partial f}{\partial u_1} = \nabla f \cdot \mathbf{U}_u
\]

Similarly, \(\frac{\partial f}{\partial u_2}\) and \(\frac{\partial f}{\partial u_3}\) can be found by taking the dot product with \(\mathbf{U}_v\) and \(\mathbf{U}_z\), respectively.

I hope this helps you find your scalar function and its partial derivatives. Let me know if you have any further questions.